A student wonders what determines the ranges of inverse trigonometric functions.
After listing the general criteria, Doctor Peterson cites several different conventions
and motivations, explaining that the definitions often come down to context and
convenience.

Find the area of a hexagon made from a triangle with squares appended to
each of its sides and three more triangles each consisting of one unknown
side and two sides which are shared with two of the squares.

Given a circle of radius r and center c, suppose two intersecting
chords AB and CD (intersecting in P) form two circular segments. How
do I compute the area of the intersection of the two circular segments?

A picture is two meters high and is hanging so the bottom of the picture
is one meter above eye level. How far from the wall should you stand so
the angle of vision occupied by the picture is a maximum?

My Dad and I are building a round pen for our horse. We have 16
16ft. panels and a 10 ft. gate and a 4ft. gate. (270 ft. total) We
want to use a radius and mark the places to dig holes for each post
that will support the panels, but we don't know how long the radius
should be. Can you help?